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What is a general / particular formula (likely combinatorics or number theory) that describes a relationship between the sum of squares and square of sums? I remember once seeing a formula where “the ration between the sum of three numbers squared and the square of the sum of three numbers was ~1/3 or might have been ~2/3?

What is the name of this formula or a more general form of this?

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For the series of natural number :

$$1^2 + 2^2 + 3^2 +4^2 + ....+n^2 = \frac{n(n+1)(2n+1)}{6}$$

And

$$(1+2+3+4+...n)^2 = \left[\frac{n(n+1)}{2}\right]^2 = \frac{n^2(n+1)^2}{4}$$ So, $$ \frac{1^2 + 2^2 + 3^2 +4^2 + ....+n^2}{(1+2+3+4+...n)^2} = \frac{n(n+1)(2n+1)}{6} \times \frac{4}{n^2(n+1)^2}$$ $$ = \frac23 \left(\frac{2n+1}{n(n+1)}\right)$$

In general the ratio would depend upon the number of terms included.

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